--- slug: most-numbers-begin-with-one passed: 2026-10-14 by: the assay line (research/assay-line/cited-1014b/), Codex fixer directed by claude-assaying-boyle-1014; source record by Codex, checked by Codex checked: 144 confirmed: 105 wrong: 38 unverifiable: 0 observed: 0 stale: 1 --- This was the directed correction pass for `most-numbers-begin-with-one` at `f5e5f84f34d7d76b07341739577895b19e12ba06`, using the 40-item triage in `research/assay-line/cited-1014b/most-numbers-begin-with-one/triage.json`. The fixer re-read each triage item's evidence before changing its authored text, data or display; the 104 CONFIRMED claims below carry the named N2 checker's source re-reads and are identified as such. Counts include 29 WRONG and nine listed MINOR claims in `wrong`, one rejected finding as CONFIRMED, and the metadata-only STALE item. This record dates the pass. The repair pass records the dek-derived share-card wording as fixed through the directed landing rebuild; the old image title and four authored links in other layers remain open pending board notes. ## Claims - WRONG | [X1] the log-circle where you multiply every number by a constant and watch the distribution refuse to move | git show d5133c8545:public/strata/most-numbers-begin-with-one/index.html | Recomputed the actual inline population sample: 67/217 leading ones at c=1, 49/217 at c=10^(-0.14), and 67/217 at c=10. A finite sample is not invariant under arbitrary rescaling. - WRONG | [X2] Most Numbers Begin With One | https://arxiv.org/pdf/1909.07527 | Example 8 gives log10(2)=0.301029995664, below one half. One has a plurality. Corrected this layer's HTML and Markdown titles, OG HTML source and election-paragraph majority claim. The PNG and WebP title and four authored links in other layers remain open under X2 below. - WRONG | [X3] street addresses of people listed in a magazine | https://pdodds.w3.uvm.edu/research/papers/others/1938/benford1938a.pdf; https://ci.nii.ac.jp/ncid/BA54818859?l=en | Read Benford printed p. 556 and the CiNii catalogue. The street addresses are from American Men of Science, a biographical directory, not a magazine. The catalogue identifies the 1933 fifth edition from Science Press. The checker's p. 558 locator was wrong. - WRONG | [X4] the numbers printed in a Reader's Digest article | https://pdodds.w3.uvm.edu/research/papers/others/1938/benford1938a.pdf | Benford printed p. 556 describes an issue of Reader's Digest, with dates and page numbers excluded, not one article. The checker's p. 558 locator was wrong. - WRONG | [X5] across all of them the leading digit landed on the same curve. | https://pdodds.w3.uvm.edu/research/papers/others/1938/benford1938a.pdf | Table I, p. 553, reports leading-one percentages of 41.3 for constants, 24.0 for specific heats and 47.2 for atomic weights. Its average of group percentages is 30.6. Corrected the universal agreement claim without calling that unweighted average a pooled frequency. - WRONG | [X6] Below are four datasets this place pulled from public sources and committed verbatim | https://physics.nist.gov/cuu/Constants/Table/allascii.txt; git show d5133c8545:research/the-first-digit/data/codata-constants.json | Read the 2022 NIST table and compared all 355 names and values. Exactly 48 JavaScript numeric values lacked source exponents in ellipsis rows. Atomic unit of action was 1.054571817 and should be 1.054571817e-34 at the printed precision; Boltzmann constant was already correct. Restored the 48 exponents in raw, compact and inline copies. Counts are unchanged. - WRONG | [X7] the populations of every country, their land areas, their economies, and the 355 physical constants in the international reference table. | https://api.worldbank.org/v2/indicator/AG.SRF.TOTL.K2?format=json; git show d5133c8545:research/the-first-digit/data/gdp-usd-2022.json; git show d5133c8545:research/the-first-digit/data/population-2022.json | The World Bank definition includes inland waters and some coastal waterways. The committed files contain 217 population, 215 surface-area and 209 GDP rows. Cuba and North Korea occur in population but not GDP; Afghanistan occurs in all three. All 355 constants remain. Coverage and surface-area labels are corrected. For the dek-derived wording, the directed disposition is: share card rebuilt at landing from the corrected dek. The current PNG and WebP still show the old pixels; neither was rendered in this repair. - WRONG | [X8] Different domains, different units, no connection between them except that each spans many orders of magnitude. | git show d5133c8545:research/the-first-digit/data/population-2022.json; git show d5133c8545:research/the-first-digit/data/gdp-usd-2022.json | Read the three raw World Bank files. They share country identifiers, including AFG for Afghanistan, so there is a connection beyond numerical ranges. Corrected the categorical independence wording without making a statistical independence claim. - WRONG | [X9] Each histogram is recomputed from the raw values in your browser, with the Benford curve laid over it as a dashed line. | git show d5133c8545:public/strata/most-numbers-begin-with-one/index.html | Read .ben and drawHist, then measured the original page in Chromium: all nine data reference dots had y=2275.765625. The content height was zero (the bounding box included a 2px border). The bars were correct. Repaired the common overlay and added a dashed polyline; browser geometry checks now test both histograms. - WRONG | [X10] Theodore Hill showed in 1995 it is the unique base-invariant one too. | https://digitalcommons.calpoly.edu/rgp_rsr/30/; https://arxiv.org/pdf/1909.07527 | Read Hill's 1995 abstract and the primer, Theorem 27 and Corollary 29. Base-invariant significand laws include mixtures with a point mass at 1; excluding that mass leaves Benford. The original uniqueness claim omitted this qualification. - WRONG | [X11] watch the dots rotate and the digit counts hold | git show d5133c8545:public/strata/most-numbers-begin-with-one/index.html | The caption promised fixed counts, but the actual sample changes from 67 to 49 leading ones at c=10^(-0.14). Corrected the caption with X1, X12 and X40; retained the rotation and recomputation. - WRONG | [X12] The dataset's numbers (each a dot at {log₁₀ value}) rotate bodily as you rescale them, yet the proportion sitting in each digit-arc stays put. | git show d5133c8545:public/strata/most-numbers-begin-with-one/index.html | renderCircle rotates fractional logarithms and recomputes a histogram of the scaled finite values. The observed population proportion changes from 30.9% to 22.6%. Corrected the sample claim while retaining ideal-law scale invariance. - MINOR | [X13] Compute as many terms as you like (exactly, with big integers) and watch the histogram settle onto the curve. | git show d5133c8545:public/strata/most-numbers-begin-with-one/index.html | Read max="4000" and NMAX=4000. The exact-integer widget offers 10 through 4000 terms. Listed MINOR for tier C as directed: this separate invitation is not changed by the overlay or limit-label repair. - WRONG | [X14] Live · leading digits of an exact integer sequence, computed with BigInt, vs Benford (dashed) | git show d5133c8545:public/strata/most-numbers-begin-with-one/index.html | The sequence panel shared the zero-content-height reference overlay; the BigInt generators themselves are exact. Fixed the overlay and retained the accurate computation/caption. - WRONG | [X15] total variation, → 0 | git show d5133c8545:public/strata/most-numbers-begin-with-one/index.html; https://arxiv.org/pdf/1909.07527 | The old label was unconditional although counting has benford:false. Recomputed counting TVD=0.2680599913 at N=500. At N=10^k-1 the first digits are uniform, not Benford, so TVD cannot tend to zero. The label is now conditional on the selected sequence. - WRONG | [X16] It needs numbers that span several orders of magnitude and are not penned in by a floor or a ceiling. | https://arxiv.org/pdf/2011.13015 | Read Hill Example 6 and Proposition 8. X=10^U for U uniform on [0,1] is Benford within [1,10]; several decades and absence of bounds are not necessary. Corrected the stated requirements and the shared data-note copies. - WRONG | [X17] assigned or bounded numbers carry no leading-digit law | https://arxiv.org/pdf/2011.13015 | Hill Example 9 gives bounded Benford laws, including a range within [73,730]. Assigned values can also have Benford first-digit probabilities. Replaced the categorical exclusion with a caveat about the actual digit distribution. - WRONG | [X18] A roulette wheel, a set of phone numbers, an invoice list rounded to the nearest \$5 | https://arxiv.org/pdf/2011.13015; git show d5133c8545:public/strata/most-numbers-begin-with-one/index.html | Choose amount 10d with probability log10(1+1/d), d=1,...,9. The amounts are multiples of five and have the exact first-digit law before and after rounding. An independent 100,000-midpoint log-uniform example rounded to five has TVD=0.0000157241447. Removed the blanket rounding exclusion. - WRONG | [X19] because the mechanism that made them never smeared them across the decades. | https://arxiv.org/pdf/2011.13015; git show d5133c8545:public/strata/most-numbers-begin-with-one/index.html | The primer's Example 10(ii) constructs Benford draws without any growth history; the assigned-invoice construction also defeats this asserted universal cause. Removed that explanation without replacing it with another universal mechanism. - WRONG | [X20] Reach for Benford only where wide, multiplicative, organically-grown numbers live. | https://arxiv.org/pdf/2011.13015 | The bounded synthetic construction in Hill Example 6 is Benford without organic growth. Replaced the only-wide-multiplicative-organic requirement with a need to justify the expected digit distribution. - MINOR | [X21] In 2011 the political scientists Deckert, Myagkov, and Ordeshook showed in Political Analysis that the first-digit test is "problematical at best as a forensic tool when applied to elections" | https://www.cambridge.org/core/services/aop-cambridge-core/content/view/3B1D64E822371C461AF3C61CE91AAF6D/S1047198700012808a.pdf/benfords-law-and-the-detection-of-election-fraud.pdf | Read Deckert, Myagkov and Ordeshook, pp. 245 and 247. The first-digit warning is supported, while the study also examines second digits. Listed MINOR for tier C as directed. The attribution sentence and its quotation are byte-for-byte unchanged. - WRONG | [X22] The datasets are real, published figures, committed as-is | https://physics.nist.gov/cuu/Constants/Table/allascii.txt; git show d5133c8545:research/the-first-digit/data/codata-constants.json | The same 48 missing CODATA exponents contradict as-is. Corrected source-fidelity wording in the colophon and README, with the raw and compact numerical repair under X6. This was an extraction error, not a stale later observation. - WRONG | [X23] Histograms on the page match the verifier exactly because rounding the snapshot to six significant figures cannot change a leading digit. | git show d5133c8545:public/strata/most-numbers-begin-with-one/index.html; git show d5133c8545:research/the-first-digit/data/page-snapshot.json | Verified all four full and compact first-digit histograms agree, before and after repair. Number((9.999999).toPrecision(6)) equals 10, changing the first digit. Constants use eight significant figures. Replaced the universal guarantee with the checked equality for these snapshots. - CONFIRMED | [X24] Last verified 19 June 2026, when it was written. | assay/lib.mjs; scripts/check-placard.mjs; assay/README.md; git show d5133c8545:research/self-check/verifier-runs.json; git show d4840e62371:public/strata/most-numbers-begin-with-one/index.html | REJECTED with evidence: assay/lib.mjs defines verified as record.passed or the writing date; scripts/check-placard.mjs and assay/README.md document a source re-read, not an automated verifier run. With no earlier assay record, 19 June 2026 was the intended hallmark. The later automated run does not falsify it. Left this generated historical line unchanged. - WRONG | [X25] and the bars track the dashed Benford curve closely (P(1) = 30.9% vs 30.1%). The data spans 5.2 orders of magnitude, which is exactly the condition the law needs. | https://arxiv.org/pdf/2011.13015; git show d5133c8545:public/strata/most-numbers-begin-with-one/index.html | Recomputed population P(1)=30.875576% and range=5.154291 decades. The numbers remain 30.9% and 5.2. Hill Proposition 8 disproves the alleged necessary/sufficient range condition. Fixed the common note and the shared overlay. - WRONG | [X26] and the bars track the dashed Benford curve closely (P(1) = 29.3% vs 30.1%). The data spans 6.2 orders of magnitude, which is exactly the condition the law needs. | https://arxiv.org/pdf/2011.13015; git show d5133c8545:public/strata/most-numbers-begin-with-one/index.html | Recomputed surface-area P(1)=29.302326% and range=6.233635 decades. The numbers remain 29.3% and 6.2. Hill Proposition 8 disproves the alleged necessary/sufficient range condition. Fixed the common note and the shared overlay. - WRONG | [X27] and the bars track the dashed Benford curve closely (P(1) = 30.1% vs 30.1%). The data spans 5.6 orders of magnitude, which is exactly the condition the law needs. | https://arxiv.org/pdf/2011.13015; git show d5133c8545:public/strata/most-numbers-begin-with-one/index.html | Recomputed GDP P(1)=30.143541% and range=5.636985 decades. The numbers remain 30.1% and 5.6. Hill Proposition 8 disproves the alleged necessary/sufficient range condition. Fixed the common note and the shared overlay. - MINOR | [X28] and it keeps shrinking. | git show d5133c8545:public/strata/most-numbers-begin-with-one/index.html | The shared convergence sentence omits fluctuations. Recomputed powers-of-two TVD increasing from 0.00388321 at N=501 to 0.00473269 at N=502. Listed MINOR for tier C; its separate sentence is unchanged. - MINOR | [X29] distance to Benford is now 0.004, and it keeps shrinking. | git show d5133c8545:public/strata/most-numbers-begin-with-one/index.html | Powers-of-two TVD at N=500 is 0.00420319, correctly displayed as 0.004. The increase from N=501 to N=502 defeats monotonic shrinking, not convergence. Listed MINOR for tier C; the shared sentence is unchanged. - MINOR | [X30] distance to Benford is now 0.001, and it keeps shrinking. | git show d5133c8545:public/strata/most-numbers-begin-with-one/index.html | Powers-of-two TVD at N=4000 is 0.00121817, correctly displayed as 0.001. The increase from N=501 to N=502 defeats monotonic shrinking, not convergence. Listed MINOR for tier C; the shared sentence is unchanged. - MINOR | [X31] distance to Benford is now 0.006, and it keeps shrinking. | git show d5133c8545:public/strata/most-numbers-begin-with-one/index.html | Fibonacci TVD at N=500 is 0.00570555, correctly displayed as 0.006, and rises to 0.00612919 at N=501. Listed MINOR for tier C as a loose convergence description; the shared sentence is unchanged. - MINOR | [X32] distance to Benford is now 0.001, and it keeps shrinking. | git show d5133c8545:public/strata/most-numbers-begin-with-one/index.html | Fibonacci TVD at N=4000 is 0.00113624, correctly displayed as 0.001. The increase from N=500 to N=501 defeats monotonic shrinking, not convergence. Listed MINOR for tier C; the shared sentence is unchanged. - MINOR | [X33] distance to Benford is now 0.047, and it keeps shrinking. | git show d5133c8545:public/strata/most-numbers-begin-with-one/index.html | Factorial TVD at N=500 is 0.04711320, correctly displayed as 0.047, and rises to 0.04808725 at N=501. Listed MINOR for tier C as a loose convergence description; the shared sentence is unchanged. - MINOR | [X34] distance to Benford is now 0.013, and it keeps shrinking. | git show d5133c8545:public/strata/most-numbers-begin-with-one/index.html | Factorial TVD at N=4000 is 0.01305194, correctly displayed as 0.013. The increase from N=500 to N=501 defeats monotonic shrinking, not convergence. Listed MINOR for tier C; the shared sentence is unchanged. - WRONG | [X35] Roger Pinkham proved in 1961 that Benford is the unique scale-invariant leading-digit law | https://dornsife.usc.edu/sergey-lototsky/wp-content/uploads/sites/211/2023/06/BenfordLaw.pdf | Read Berger and Hill section 4.2, pp. 40-41: Pinkham's argument assumes an impossible scale-invariant probability on all positive reals. Their Theorem 4.20, pp. 42-43, gives a valid significant-digit formulation and proof. Corrected the proof attribution and bibliography gloss while preserving uniqueness. - WRONG | [X36] Both are what you get when quantities are spread evenly across orders of magnitude rather than added up. | https://arxiv.org/pdf/cond-mat/0412004 | Read Newman equation (4) and Appendix A. Density and rank/cumulative exponents differ by one. An inverse-rank list has ten times as many entries in successive lower-frequency decades, unlike equal mass per logarithmic interval. Corrected the related explanation and its visible summary. - WRONG | [X37] a first digit quietly encodes whether a dataset grew multiplicatively. | https://arxiv.org/pdf/1909.07527 | The primer's Theorem 16 characterizes uniform fractional logarithms, not growth history. Example 10(ii) generates Benford draws directly as 10^U. Corrected both related-note copies; a first-digit pattern cannot identify the generating history. - STALE | [X38] "dateModified":"2026-06-19" | git log d5133c8545 -- public/strata/most-numbers-begin-with-one/index.html; git show d4840e62371:public/strata/most-numbers-begin-with-one/index.html | The original dateModified was 2026-06-19 despite subsequent substantive placard additions in the pinned history. This substantive repair sets it to 2026-10-14 and keeps datePublished=2026-06-19. The user's metadata exception requires no Updated or Corrected line for this field. - WRONG | [X39] Live · leading digits of a real dataset vs Benford (dashed) | git show d5133c8545:public/strata/most-numbers-begin-with-one/index.html | The data histogram uses the same failed .ben layout as X9 and X14. The old nine reference marks shared one y coordinate. The repaired overlay uses the bar plotting area and a dashed polyline, with geometry checked at desktop and mobile widths. - WRONG | [X40] (added) barely moves | git show d5133c8545:public/strata/most-numbers-begin-with-one/index.html | Read the visible barely moves sublabel and independently swept all 4001 slider settings. Population P(1) spans 22.6%-35.5%; constants span 22.5%-37.2%. Replaced the characterization with sample proportion varies, retaining the computed readout. - CONFIRMED | In many real-world lists of numbers, the first digit is a 1 about 30% of the time and a 9 less than 5% of the time. | https://arxiv.org/pdf/1909.07527 | Carried from the N2 Codex checker's source re-read: [001 plain-frequency] Read Example 8 in the fetched PDF, p. 3: the limit is log(1+1/d), with decimal log. Independently evaluated P(1)=0.301029995664 and P(9)=0.045757490561. The four committed datasets provide concrete examples. The claim is approximate, not a statement about every arbitrary list. - CONFIRMED | the page proves why it happens and shows exactly where the popular 'the numbers break Benford, so it must be fraud' claim goes wrong. | git show d5133c8545:public/strata/most-numbers-begin-with-one/index.html | Carried from the N2 Codex checker's source re-read: [002 plain-page-proof] Read sections I-IV and the code: four empirical histograms, exact integer sequences, the rotation argument, and the warning against treating nonconformity as a fraud verdict are present. The empirical displays illustrate the law; the mathematical proof is conditional on scale invariance. Specific false boundaries and display claims are assessed separately. - CONFIRMED | A 1 turns up about 30% of the time; a 9, less than 5%. | https://arxiv.org/pdf/1909.07527 | Carried from the N2 Codex checker's source re-read: [003 deck-frequency] Read Example 8 in the fetched PDF, p. 3: the limit is log(1+1/d), with decimal log. Independently evaluated P(1)=0.301029995664 and P(9)=0.045757490561. The four committed datasets provide concrete examples. The claim is approximate, not a statement about every arbitrary list. - CONFIRMED | That lopsidedness is Benford's law, P(d) = log₁₀(1 + 1/d) | https://arxiv.org/pdf/1909.07527 | Carried from the N2 Codex checker's source re-read: [004 deck-law] Read Example 8 in the fetched PDF, p. 3: the limit is log(1+1/d), with decimal log. Independently evaluated P(1)=0.301029995664 and P(9)=0.045757490561. The four committed datasets provide concrete examples. The claim is approximate, not a statement about every arbitrary list. - CONFIRMED | it is the only leading-digit pattern that survives a change of units. | https://arxiv.org/pdf/1909.07527 | Carried from the N2 Codex checker's source re-read: [005 deck-unique] Theorem 21: "A random variable X with P (X = 0) = 0 is Benford if and only if it has scale-invariant significant digits." Theorem 23 requires invariance for all positive scale factors. Read the page in its universal-law context, not as an assertion that one particular conversion suffices. - CONFIRMED | An immersive page proves it three ways (on four real datasets, on the powers of two by exact integer arithmetic, and on the log-circle where scale-invariance forces it) | git show d5133c8545:public/strata/most-numbers-begin-with-one/index.html | Carried from the N2 Codex checker's source re-read: [006 deck-three-proofs] Read sections I-IV and the code: four empirical histograms, exact integer sequences, the rotation argument, and the warning against treating nonconformity as a fraud verdict are present. The empirical displays illustrate the law; the mathematical proof is conditional on scale invariance. Specific false boundaries and display claims are assessed separately. - CONFIRMED | The astronomer Simon Newcomb spotted it in 1881 from the way logarithm books wore out at the front | https://pdodds.w3.uvm.edu/research/papers/others/1881/newcomb1881a.pdf | Carried from the N2 Codex checker's source re-read: [008 intro-newcomb] Read both pages of Newcomb and the 1881 cover: "noticing how much faster the first pages wear out than the last ones" and "The first significant figure is oftener 1 than any other digit". Publication year is 1881; the occupation is independently supported by the Princeton chapter. - CONFIRMED | the physicist Frank Benford counted 20,229 numbers in 1938 and made it a law | https://pdodds.w3.uvm.edu/research/papers/others/1938/benford1938a.pdf | Carried from the N2 Codex checker's source re-read: [009 intro-benford] Read Benford pp. 551-553 and Table I. The heading says "20,229 OBSERVATIONS"; twenty rows A-T include "Rivers, Area", "Drainage", and "Atomic Wgt.". The publication is March 1938. Treat the date as the publication date, not a dated lab record of when every observation was collected. - CONFIRMED | P(d) = log₁₀(1 + 1/d) | https://arxiv.org/pdf/1909.07527 | Carried from the N2 Codex checker's source re-read: [010 intro-law] Read Example 8 in the fetched PDF, p. 3: the limit is log(1+1/d), with decimal log. Independently evaluated P(1)=0.301029995664 and P(9)=0.045757490561. The four committed datasets provide concrete examples. The claim is approximate, not a statement about every arbitrary list. - CONFIRMED | the law is the **only** leading-digit pattern that doesn't change when you switch units | https://arxiv.org/pdf/1909.07527 | Carried from the N2 Codex checker's source re-read: [011 intro-unique] Theorem 21: "A random variable X with P (X = 0) = 0 is Benford if and only if it has scale-invariant significant digits." Theorem 23 requires invariance for all positive scale factors. Read the page in its universal-law context, not as an assertion that one particular conversion suffices. - CONFIRMED | "a law of the world shouldn't care which country writes the river down" is enough, by itself, to force the whole curve. | https://arxiv.org/pdf/1909.07527 | Carried from the N2 Codex checker's source re-read: [012 intro-units-force] Theorem 21: "A random variable X with P (X = 0) = 0 is Benford if and only if it has scale-invariant significant digits." Theorem 23 requires invariance for all positive scale factors. Read the page in its universal-law context, not as an assertion that one particular conversion suffices. - CONFIRMED | the law emerging from pure arithmetic on the powers of two, the Fibonacci numbers and the factorials, computed exactly with big integers | git show d5133c8545:public/strata/most-numbers-begin-with-one/index.html | Carried from the N2 Codex checker's source re-read: [014 intro-exact-sequences] Read and executed the pinned inline script locally. The code uses "let x = 1n", "let a = 1n, b = 1n", "let f = 1n", then "Number(x.toString()[0])" and the corresponding Fibonacci/factorial leading characters. The law uses "log10(1 + 1 / d)". - CONFIRMED | where forensic accountants legitimately use it as a *filter* | https://www.journalofaccountancy.com/issues/2022/sep/using-benfords-law-reveal-journal-entry-irregularities/ | Carried from the N2 Codex checker's source re-read: [016 intro-filter] The fetched Nigrini article says testing "could direct auditors" to entries below a threshold, "which could be an indicator" of fraud. This supports a screening lead, not a conclusive fraud verdict. - CONFIRMED | as the political scientists who tested it put it, *"problematical at best."* | https://www.cambridge.org/core/journals/political-analysis/article/benfords-law-and-the-detection-of-election-fraud/3B1D64E822371C461AF3C61CE91AAF6D | Carried from the N2 Codex checker's source re-read: [017 intro-deckert-quote] The words "problematical at best" are in the abstract. The period inside the page quotation closes the excerpted sentence; the tool warning is a punctuation-boundary false alarm, not an altered quotation. - CONFIRMED | In many real-world lists, the first digit is a 1 about 30% of the time and a 9 less than 5%. | https://arxiv.org/pdf/1909.07527 | Carried from the N2 Codex checker's source re-read: [021 meta-world-list] Read Example 8 in the fetched PDF, p. 3: the limit is log(1+1/d), with decimal log. Independently evaluated P(1)=0.301029995664 and P(9)=0.045757490561. The four committed datasets provide concrete examples. The claim is approximate, not a statement about every arbitrary list. This carried confirmation concerns the frequency clause; X7 corrects the incidental land-area label. - CONFIRMED | A 1 shows up about thirty percent of the time. A 9, less than five. | https://arxiv.org/pdf/1909.07527 | Carried from the N2 Codex checker's source re-read: [022 frame-frequency] Read Example 8 in the fetched PDF, p. 3: the limit is log(1+1/d), with decimal log. Independently evaluated P(1)=0.301029995664 and P(9)=0.045757490561. The four committed datasets provide concrete examples. The claim is approximate, not a statement about every arbitrary list. - CONFIRMED | People looked up numbers beginning with 1 far more often than numbers beginning with 9 | https://pdodds.w3.uvm.edu/research/papers/others/1881/newcomb1881a.pdf | Carried from the N2 Codex checker's source re-read: [025 newcomb-wear] Read both pages of Newcomb and the 1881 cover: "noticing how much faster the first pages wear out than the last ones" and "The first significant figure is oftener 1 than any other digit". Publication year is 1881; the occupation is independently supported by the Princeton chapter. - CONFIRMED | so the early pages wore out first. | https://pdodds.w3.uvm.edu/research/papers/others/1881/newcomb1881a.pdf | Carried from the N2 Codex checker's source re-read: [026 newcomb-early-pages] Read both pages of Newcomb and the 1881 cover: "noticing how much faster the first pages wear out than the last ones" and "The first significant figure is oftener 1 than any other digit". Publication year is 1881; the occupation is independently supported by the Princeton chapter. - CONFIRMED | in a table of "natural numbers," | https://pdodds.w3.uvm.edu/research/papers/others/1881/newcomb1881a.pdf | Carried from the N2 Codex checker's source re-read: [027 newcomb-natural-quote] Newcomb writes "As natural numbers occur in nature". The page quotes only the term "natural numbers"; placing its comma inside quotation marks changes punctuation, not the quoted term. Declined the automated literal-comma warning. - CONFIRMED | the leading digit is a 1 about 30% of the time, and the frequency falls steadily to the 9s. | https://pdodds.w3.uvm.edu/research/papers/others/1881/newcomb1881a.pdf; https://press.princeton.edu/chapters/s10527.pdf | Carried from the N2 Codex checker's source re-read: [028 newcomb-frequency] Newcomb says "the frequency diminishes up to 9". The record excerpt alone omits the numerical table; the Princeton chapter reproduces Newcomb Table 1.1 with first-digit probabilities 0.3010 for 1 and 0.0458 for 9. - CONFIRMED | He published two pages on it | https://pdodds.w3.uvm.edu/research/papers/others/1881/newcomb1881a.pdf | Carried from the N2 Codex checker's source re-read: [029 newcomb-two-pages] The fetched bibliographic cover reads "Simon Newcomb", "American Journal of Mathematics, Vol. 4, No. 1. (1881), pp. 39-40." Both article pages are present. The online DOI check for 10.2307/2369148 returned MATCH. - CONFIRMED | the world forgot. | https://dornsife.usc.edu/sergey-lototsky/wp-content/uploads/sites/211/2023/06/BenfordLaw.pdf | Carried from the N2 Codex checker's source re-read: [030 newcomb-forgotten] Berger and Hill explicitly describe "Newcomb’s article having been forgotten at the time" and the result coming to be known as Benford’s Law. The historical priority and later eponym are supported. - CONFIRMED | Fifty-seven years later the physicist Frank Benford rediscovered the same wear pattern | https://dornsife.usc.edu/sergey-lototsky/wp-content/uploads/sites/211/2023/06/BenfordLaw.pdf; https://pdodds.w3.uvm.edu/research/papers/others/1938/benford1938a.pdf | Carried from the N2 Codex checker's source re-read: [031 benford-rediscovery] Berger and Hill describe rediscovery by Benford; the primary 1938 paper discusses the worn tables. The publication years differ by 1938-1881=57. This checks the conventional publication-year chronology, not an exact date for Benford noticing wear. - CONFIRMED | the physicist Frank Benford | https://pdodds.w3.uvm.edu/research/papers/others/1938/benford1938a.pdf | Carried from the N2 Codex checker's source re-read: [032 benford-physicist] The author affiliation is "Physicist, Research Laboratory, General Electric Company". - CONFIRMED | did the work Newcomb hadn't: he counted. | https://pdodds.w3.uvm.edu/research/papers/others/1881/newcomb1881a.pdf; https://pdodds.w3.uvm.edu/research/papers/others/1938/benford1938a.pdf | Carried from the N2 Codex checker's source re-read: [033 benford-newcomb-not-count] Read all of Newcomb pp. 39-40: an argument and a table of calculated probabilities, not a reported empirical collection. Positive control: the probability table is present. Benford Table I explicitly reports observed counts for twenty groups. This supports the contrast between these two papers, not a claim about everything Newcomb ever did. - CONFIRMED | 20,229 numbers, gathered from twenty unrelated tables | https://pdodds.w3.uvm.edu/research/papers/others/1938/benford1938a.pdf | Carried from the N2 Codex checker's source re-read: [034 benford-20229] Read Benford pp. 551-553 and Table I. The heading says "20,229 OBSERVATIONS"; twenty rows A-T include "Rivers, Area", "Drainage", and "Atomic Wgt.". The publication is March 1938. Treat the date as the publication date, not a dated lab record of when every observation was collected. - CONFIRMED | river drainage areas, atomic weights | https://pdodds.w3.uvm.edu/research/papers/others/1938/benford1938a.pdf | Carried from the N2 Codex checker's source re-read: [035 benford-rivers-atomic] Read Benford pp. 551-553 and Table I. The heading says "20,229 OBSERVATIONS"; twenty rows A-T include "Rivers, Area", "Drainage", and "Atomic Wgt.". The publication is March 1938. Treat the date as the publication date, not a dated lab record of when every observation was collected. - CONFIRMED | The law now carries his name, though Newcomb had it first | https://dornsife.usc.edu/sergey-lototsky/wp-content/uploads/sites/211/2023/06/BenfordLaw.pdf | Carried from the N2 Codex checker's source re-read: [039 benford-name] Berger and Hill explicitly describe "Newcomb’s article having been forgotten at the time" and the result coming to be known as Benford’s Law. The historical priority and later eponym are supported. - CONFIRMED | a law named after the wrong person | https://dornsife.usc.edu/sergey-lototsky/wp-content/uploads/sites/211/2023/06/BenfordLaw.pdf | Carried from the N2 Codex checker's source re-read: [040 stigler] Berger and Hill explicitly describe "Newcomb’s article having been forgotten at the time" and the result coming to be known as Benford’s Law. The historical priority and later eponym are supported. - CONFIRMED | The law · P(d) = log₁₀(1 + 1/d), computed in your browser | git show d5133c8545:public/strata/most-numbers-begin-with-one/index.html | Carried from the N2 Codex checker's source re-read: [041 law-browser] Read and executed the pinned inline script locally. The code uses "let x = 1n", "let a = 1n, b = 1n", "let f = 1n", then "Number(x.toString()[0])" and the corresponding Fibonacci/factorial leading characters. The law uses "log10(1 + 1 / d)". - CONFIRMED | a leading 1 is about six times more likely than a leading 9. | https://arxiv.org/pdf/1909.07527 | Carried from the N2 Codex checker's source re-read: [042 six-times] Independently evaluated the formula: P(1)/P(9)=6.57881348, supporting the rough "about six"; sum(d=5..9) log10((d+1)/d)=log10(10/5)=log10(2)=0.301029995664. The equality is exact; the percentage is rounded. - CONFIRMED | 1 occurs precisely as often as 5, 6, 7, 8 and 9 put together. | https://arxiv.org/pdf/1909.07527 | Carried from the N2 Codex checker's source re-read: [043 one-equals-five-nine] Independently evaluated the formula: P(1)/P(9)=6.57881348, supporting the rough "about six"; sum(d=5..9) log10((d+1)/d)=log10(10/5)=log10(2)=0.301029995664. The equality is exact; the percentage is rounded. - CONFIRMED | Both equal log₁₀2 = 30.103% | https://arxiv.org/pdf/1909.07527 | Carried from the N2 Codex checker's source re-read: [044 log2-30103] Independently evaluated the formula: P(1)/P(9)=6.57881348, supporting the rough "about six"; sum(d=5..9) log10((d+1)/d)=log10(10/5)=log10(2)=0.301029995664. The equality is exact; the percentage is rounded. - CONFIRMED | the sum P(5)+…+P(9) telescopes to log₁₀(10/5) = log₁₀2. | https://arxiv.org/pdf/1909.07527 | Carried from the N2 Codex checker's source re-read: [045 telescopes] Independently evaluated the formula: P(1)/P(9)=6.57881348, supporting the rough "about six"; sum(d=5..9) log10((d+1)/d)=log10(10/5)=log10(2)=0.301029995664. The equality is exact; the percentage is rounded. - CONFIRMED | conforms @5% if < 15.51 | https://www.itl.nist.gov/div898/handbook/eda/section3/eda3674.htm | Carried from the N2 Codex checker's source re-read: [050 chi-threshold] NIST upper-tail table has row 8: "13.362 15.507 17.535 20.090 26.125". The 0.95 quantile is 15.507, rounded to 15.51; 9 digit bins with a specified distribution give 8 degrees of freedom. - CONFIRMED | Switching units multiplies every number in the list by the same constant | https://arxiv.org/pdf/1909.07527 | Carried from the N2 Codex checker's source re-read: [051 unit-multiplication] For the listed ratio-scale examples, miles/kilometres and dollars/yen, changing units multiplies values by a positive conversion factor. The primer defines scale-invariant digits through aX. The page is not discussing affine temperature conversions. - CONFIRMED | is enough to pin down the law completely. | https://arxiv.org/pdf/1909.07527 | Carried from the N2 Codex checker's source re-read: [052 scale-sufficient] Theorem 21: "A random variable X with P (X = 0) = 0 is Benford if and only if it has scale-invariant significant digits." Theorem 23 requires invariance for all positive scale factors. Read the page in its universal-law context, not as an assertion that one particular conversion suffices. - CONFIRMED | Lay the numbers on a logarithmic circle, where a value's position is the fractional part of its base-10 logarithm. | https://arxiv.org/pdf/1909.07527 | Carried from the N2 Codex checker's source re-read: [053 circle-position] Theorem 16 says the fractional logarithm is "uniformly distributed on [0, 1]" exactly when X is Benford. This verifies the log-circle construction. The source record locates this theorem at p. 5, but it is on printed/PDF p. 6. - CONFIRMED | A number leads with digit d exactly when it lands in the arc from log₁₀d to log₁₀(d+1) | https://arxiv.org/pdf/1909.07527 | Carried from the N2 Codex checker's source re-read: [054 circle-digit-arc] Independently derived from x=10^k s, 1<=s<10: first digit d iff d<=s`. No Updated line was added: X38 is the explicit JSON-LD metadata exception. [public/strata/most-numbers-begin-with-one/index.html:425](/home/liam/aw-claude-assaying-boyle-1014/public/strata/most-numbers-begin-with-one/index.html:425); [src/content/strata/most-numbers-begin-with-one.md:47](/home/liam/aw-claude-assaying-boyle-1014/src/content/strata/most-numbers-begin-with-one.md:47) Corrected 2026-10-14: The title said most numbers begin with one. Under Benford's law, one leads about 30.1% of the time, the largest share for any digit, not a majority; found by the assay line, 2026-10-14. [public/strata/most-numbers-begin-with-one/index.html:426](/home/liam/aw-claude-assaying-boyle-1014/public/strata/most-numbers-begin-with-one/index.html:426); [src/content/strata/most-numbers-begin-with-one.md:49](/home/liam/aw-claude-assaying-boyle-1014/src/content/strata/most-numbers-begin-with-one.md:49) Corrected 2026-10-14: The page placed Benford's street addresses in a magazine, his Reader's Digest sample in one article, and all twenty tables on the same curve. The addresses came from the biographical directory American Men of Science; he counted an issue of Reader's Digest excluding dates and page numbers; the average of the tables' percentages followed the curve closely, with differing individual fits; found by the assay line, 2026-10-14. [public/strata/most-numbers-begin-with-one/index.html:427](/home/liam/aw-claude-assaying-boyle-1014/public/strata/most-numbers-begin-with-one/index.html:427); [src/content/strata/most-numbers-begin-with-one.md:51](/home/liam/aw-claude-assaying-boyle-1014/src/content/strata/most-numbers-begin-with-one.md:51) Corrected 2026-10-14: The page called its datasets verbatim and as-is. They are numerical extractions, and 48 CODATA values had lost exponents. Those exponents are restored from the 2022 NIST table; the first-digit counts are unchanged; found by the assay line, 2026-10-14. [public/strata/most-numbers-begin-with-one/index.html:428](/home/liam/aw-claude-assaying-boyle-1014/public/strata/most-numbers-begin-with-one/index.html:428); [src/content/strata/most-numbers-begin-with-one.md:53](/home/liam/aw-claude-assaying-boyle-1014/src/content/strata/most-numbers-begin-with-one.md:53) Corrected 2026-10-14: The page described every country, land areas, and datasets with no other connection. The World Bank snapshots contain 217 population, 215 surface-area and 209 GDP entries for overlapping sets of countries and economies; surface area includes water; found by the assay line, 2026-10-14. [public/strata/most-numbers-begin-with-one/index.html:429](/home/liam/aw-claude-assaying-boyle-1014/public/strata/most-numbers-begin-with-one/index.html:429); [src/content/strata/most-numbers-begin-with-one.md:55](/home/liam/aw-claude-assaying-boyle-1014/src/content/strata/most-numbers-begin-with-one.md:55) Corrected 2026-10-14: The page said the sample digit counts held, stayed put or barely moved under rescaling. The population snapshot's leading-one share changes from 30.9% to 22.6% when multiplied by 10^(-0.14). Scale invariance describes the ideal Benford law; finite-sample proportions can change; found by the assay line, 2026-10-14. [public/strata/most-numbers-begin-with-one/index.html:430](/home/liam/aw-claude-assaying-boyle-1014/public/strata/most-numbers-begin-with-one/index.html:430); [src/content/strata/most-numbers-begin-with-one.md:57](/home/liam/aw-claude-assaying-boyle-1014/src/content/strata/most-numbers-begin-with-one.md:57) Corrected 2026-10-14: The page promised dashed Benford comparisons, but the reference marks collapsed to a common level. The data and sequence histograms now draw a dashed curve at the computed Benford proportions; found by the assay line, 2026-10-14. [public/strata/most-numbers-begin-with-one/index.html:431](/home/liam/aw-claude-assaying-boyle-1014/public/strata/most-numbers-begin-with-one/index.html:431); [src/content/strata/most-numbers-begin-with-one.md:59](/home/liam/aw-claude-assaying-boyle-1014/src/content/strata/most-numbers-begin-with-one.md:59) Corrected 2026-10-14: The page credited Pinkham's 1961 argument as a proof. That argument is flawed; scale-invariance uniqueness holds for significant-digit distributions, as proved in Berger and Hill's Theorem 4.20; found by the assay line, 2026-10-14. [public/strata/most-numbers-begin-with-one/index.html:432](/home/liam/aw-claude-assaying-boyle-1014/public/strata/most-numbers-begin-with-one/index.html:432); [src/content/strata/most-numbers-begin-with-one.md:61](/home/liam/aw-claude-assaying-boyle-1014/src/content/strata/most-numbers-begin-with-one.md:61) Corrected 2026-10-14: The page said base invariance alone uniquely selects Benford. Base-invariant significand laws can mix Benford with a point mass at 1; excluding that mass restores uniqueness; found by the assay line, 2026-10-14. [public/strata/most-numbers-begin-with-one/index.html:433](/home/liam/aw-claude-assaying-boyle-1014/public/strata/most-numbers-begin-with-one/index.html:433); [src/content/strata/most-numbers-begin-with-one.md:63](/home/liam/aw-claude-assaying-boyle-1014/src/content/strata/most-numbers-begin-with-one.md:63) Corrected 2026-10-14: The counting-number display labelled its distance to Benford as tending to zero. Its first-digit proportions do not converge to Benford; the limit label now follows the selected sequence; found by the assay line, 2026-10-14. [public/strata/most-numbers-begin-with-one/index.html:434](/home/liam/aw-claude-assaying-boyle-1014/public/strata/most-numbers-begin-with-one/index.html:434); [src/content/strata/most-numbers-begin-with-one.md:65](/home/liam/aw-claude-assaying-boyle-1014/src/content/strata/most-numbers-begin-with-one.md:65) Corrected 2026-10-14: The page required several decades and organic multiplicative growth, and ruled out bounded, assigned or rounded data. These are not necessary conditions: X = 10^U for uniform U on [0,1] is bounded and Benford, and amounts already divisible by five can follow the first-digit law without rounding changing them; found by the assay line, 2026-10-14. [public/strata/most-numbers-begin-with-one/index.html:435](/home/liam/aw-claude-assaying-boyle-1014/public/strata/most-numbers-begin-with-one/index.html:435); [src/content/strata/most-numbers-begin-with-one.md:67](/home/liam/aw-claude-assaying-boyle-1014/src/content/strata/most-numbers-begin-with-one.md:67) Corrected 2026-10-14: The page said rounding to six significant figures cannot change a leading digit. It can: 9.999999 rounds to 10. The compacted snapshot's histograms have instead been checked against the full snapshots; found by the assay line, 2026-10-14. [public/strata/most-numbers-begin-with-one/index.html:436](/home/liam/aw-claude-assaying-boyle-1014/public/strata/most-numbers-begin-with-one/index.html:436); [src/content/strata/most-numbers-begin-with-one.md:69](/home/liam/aw-claude-assaying-boyle-1014/src/content/strata/most-numbers-begin-with-one.md:69) Corrected 2026-10-14: The related note said Zipf and Benford both come from equal spread across logarithmic intervals. Zipf's inverse-rank relation and Benford's uniform fractional logarithms describe different distributions; found by the assay line, 2026-10-14. [public/strata/most-numbers-begin-with-one/index.html:437](/home/liam/aw-claude-assaying-boyle-1014/public/strata/most-numbers-begin-with-one/index.html:437); [src/content/strata/most-numbers-begin-with-one.md:71](/home/liam/aw-claude-assaying-boyle-1014/src/content/strata/most-numbers-begin-with-one.md:71) Corrected 2026-10-14: The related note said a first digit reveals whether a dataset grew multiplicatively. A leading-digit fit does not establish how the dataset was generated; found by the assay line, 2026-10-14. ## Edit ledger All locations below refer to the final working files; the diff hunk headers give both old and new line numbers. Before-lines begin with `-`, after-lines with `+`. Existing em dashes are transcribed as `\u2014`. The DATA statement and two one-line JSON datasets are itemized by named constant below instead of reproducing thousands of unrelated unchanged values. - [public/strata/most-numbers-begin-with-one/index.html:6](/home/liam/aw-claude-assaying-boyle-1014/public/strata/most-numbers-begin-with-one/index.html:6): before and after: ```diff @@ -6,2 +6,2 @@ -Benford's Law: Why Most Numbers Begin With 1 - +Benford's Law: One Leads Most Often + ``` - [public/strata/most-numbers-begin-with-one/index.html:65](/home/liam/aw-claude-assaying-boyle-1014/public/strata/most-numbers-begin-with-one/index.html:65): before and after: ```diff @@ -65,1 +65,2 @@ - .ben { position: absolute; left: 0; right: 0; height: 0; border-top: 2px dashed var(--teal); z-index: 3; } + .ben { position: absolute; inset: 0; pointer-events: none; z-index: 3; } + .ben svg { position: absolute; inset: 0; width: 100%; height: 100%; overflow: visible; } ``` - [public/strata/most-numbers-begin-with-one/index.html:115](/home/liam/aw-claude-assaying-boyle-1014/public/strata/most-numbers-begin-with-one/index.html:115): before and after: ```diff @@ -114,1 +115,1 @@ - + ``` - [public/strata/most-numbers-begin-with-one/index.html:119](/home/liam/aw-claude-assaying-boyle-1014/public/strata/most-numbers-begin-with-one/index.html:119): before and after: ```diff @@ -118,1 +119,1 @@ - + ``` - [public/strata/most-numbers-begin-with-one/index.html:128](/home/liam/aw-claude-assaying-boyle-1014/public/strata/most-numbers-begin-with-one/index.html:128): before and after: ```diff @@ -127,1 +128,1 @@ -

Most Numbers Begin With One

+

Benford's Law: One Leads Most Often

``` - [public/strata/most-numbers-begin-with-one/index.html:137](/home/liam/aw-claude-assaying-boyle-1014/public/strata/most-numbers-begin-with-one/index.html:137): before and after: ```diff @@ -136,1 +137,1 @@ -

Fifty-seven years later the physicist Frank Benford rediscovered the same wear pattern and did the work Newcomb hadn't: he counted. 20,229 numbers, gathered from twenty unrelated tables \u2014 river drainage areas, atomic weights, street addresses of people listed in a magazine, the numbers printed in a Reader's Digest article \u2014 and across all of them the leading digit landed on the same curve. The law now carries his name, though Newcomb had it first; it is, fittingly, an example of a law named after the wrong person.

+

Fifty-seven years later the physicist Frank Benford rediscovered the same wear pattern and did the work Newcomb hadn't: he counted. 20,229 numbers, gathered from twenty tables: river drainage areas, atomic weights, street addresses from the biographical directory American Men of Science, the numbers in an issue of Reader's Digest (excluding dates and page numbers). The average of the twenty tables' first-digit percentages followed the logarithmic curve closely, though individual tables differed substantially. The law now carries his name, though Newcomb had it first; it is, fittingly, an example of a law named after the wrong person.

``` - [public/strata/most-numbers-begin-with-one/index.html:157](/home/liam/aw-claude-assaying-boyle-1014/public/strata/most-numbers-begin-with-one/index.html:157): before and after: ```diff @@ -156,1 +157,1 @@ -

A formula is one thing; the world is another. Below are four datasets this place pulled from public sources and committed verbatim \u2014 the populations of every country, their land areas, their economies, and the 355 physical constants in the international reference table. Different domains, different units, no connection between them except that each spans many orders of magnitude. Each histogram is recomputed from the raw values in your browser, with the Benford curve laid over it as a dashed line.

+

A formula is one thing; the world is another. Below are four numerical snapshots extracted from public sources: population entries for 217 countries and economies, surface-area entries for 215, GDP entries for 209, and the 355 physical constants in the international reference table. The first three measure overlapping sets of countries and economies; surface area includes inland waters and some coastal waterways. These are different measurements in different units, each spanning many orders of magnitude. Each histogram is recomputed from the compacted values in your browser, with the Benford curve laid over it as a dashed line.

``` - [public/strata/most-numbers-begin-with-one/index.html:181](/home/liam/aw-claude-assaying-boyle-1014/public/strata/most-numbers-begin-with-one/index.html:181): before and after: ```diff @@ -180,1 +181,1 @@ -

That single demand \u2014 scale invariance \u2014 is enough to pin down the law completely. Lay the numbers on a logarithmic circle, where a value's position is the fractional part of its base-10 logarithm. A number leads with digit d exactly when it lands in the arc from log₁₀d to log₁₀(d+1) \u2014 and that arc has length log₁₀(1+1/d), the Benford probability, by construction. Multiplying every number by a constant rotates the whole circle. The dots move; the arc lengths do not. So the only distribution that looks the same after any rotation is the one that's uniform around the circle \u2014 and uniform-on-the-log-circle is Benford's law. Roger Pinkham proved in 1961 that Benford is the unique scale-invariant leading-digit law; Theodore Hill showed in 1995 it is the unique base-invariant one too.

+

That single demand \u2014 scale invariance \u2014 is enough to pin down the law completely. Lay the numbers on a logarithmic circle, where a value's position is the fractional part of its base-10 logarithm. A number leads with digit d exactly when it lands in the arc from log₁₀d to log₁₀(d+1) \u2014 and that arc has length log₁₀(1+1/d), the Benford probability, by construction. Multiplying every number by a constant rotates the whole circle. The dots move; the arc lengths do not. So the only distribution that looks the same after any rotation is the one that's uniform around the circle \u2014 and uniform-on-the-log-circle is Benford's law. Roger Pinkham argued for this in 1961, but his proof was flawed. The uniqueness statement holds for significant-digit distributions; Berger and Hill give a valid proof in Theorem 4.20. Theodore Hill's 1995 base-invariance result allows mixtures of Benford with a point mass at significand 1. Benford is unique when that point mass is excluded.

``` - [public/strata/most-numbers-begin-with-one/index.html:185](/home/liam/aw-claude-assaying-boyle-1014/public/strata/most-numbers-begin-with-one/index.html:185): before and after: ```diff @@ -184,1 +185,1 @@ -
Live · the log-circle. Drag to multiply the dataset by a constant \u2014 watch the dots rotate and the digit counts hold
+
Live · the log-circle. Drag to multiply the dataset by a constant: watch the dots rotate and the sample digit counts change
``` - [public/strata/most-numbers-begin-with-one/index.html:191](/home/liam/aw-claude-assaying-boyle-1014/public/strata/most-numbers-begin-with-one/index.html:191): before and after: ```diff @@ -190,1 +191,1 @@ -
P(1) after ×c
\u2014
barely moves
+
P(1) after ×c
\u2014
sample proportion varies
``` - [public/strata/most-numbers-begin-with-one/index.html:201](/home/liam/aw-claude-assaying-boyle-1014/public/strata/most-numbers-begin-with-one/index.html:201): before and after: ```diff @@ -200,1 +201,1 @@ -

The dataset's numbers (each a dot at {log₁₀ value}) rotate bodily as you rescale them, yet the proportion sitting in each digit-arc stays put. Change the units and the law doesn't budge. That invariance is not a happy accident of this dataset \u2014 it is the property that defines the law, and the reason it turns up everywhere at once.

+

The dataset's numbers (each a dot at {log₁₀ value}) rotate bodily as you rescale them, and the sample proportion in each digit-arc can change. For the population snapshot, P(1) changes from 30.9% at c = 1 to 22.6% at c = 10−0.14. The ideal Benford probability law is scale-invariant. That does not make a finite sample's digit counts invariant.

``` - [public/strata/most-numbers-begin-with-one/index.html:223](/home/liam/aw-claude-assaying-boyle-1014/public/strata/most-numbers-begin-with-one/index.html:223): before and after: ```diff @@ -222,1 +223,1 @@ -
distance to Benford
\u2014
total variation, → 0
+
distance to Benford
\u2014
total variation, → 0
``` - [public/strata/most-numbers-begin-with-one/index.html:231](/home/liam/aw-claude-assaying-boyle-1014/public/strata/most-numbers-begin-with-one/index.html:231): before and after: ```diff @@ -230,1 +231,1 @@ -

Benford is not a law of nature, and the honest half of the story is its boundary. It needs numbers that span several orders of magnitude and are not penned in by a floor or a ceiling. Adult human heights in centimetres are nearly all 1-something \u2014 no spread, no Benford. A roulette wheel, a set of phone numbers, an invoice list rounded to the nearest \$5 \u2014 assigned or bounded numbers carry no leading-digit law, because the mechanism that made them never smeared them across the decades. Even the physical constants above lead with 1 a little too often (35.5% against the expected 30.1%); they still pass a χ² test, but the fit is visibly looser than the populations'. Reach for Benford only where wide, multiplicative, organically-grown numbers live.

+

Benford is not a law of nature, and the honest half of the story is its boundary. A wide range alone does not guarantee Benford, and several decades are not necessary. Draw U uniformly from [0,1] and set X = 10U: the result is Benford, bounded between 1 and 10, with no growth history required. Adult human heights in centimetres are nearly all 1-something, so that distribution is far from Benford. Bounds, assigned values and rounding can affect first digits, but none alone decides whether their distribution fits. Even amounts rounded to the nearest $5 can follow the first-digit law: choose 10d with probability log₁₀(1 + 1/d), for d = 1,…,9; every amount is already a multiple of five. Even the physical constants above lead with 1 a little too often (35.5% against the expected 30.1%); they still pass a χ² test, but the fit is visibly looser than the populations'. Use Benford only when there is a reason to expect its digit distribution in the data at hand.

``` - [public/strata/most-numbers-begin-with-one/index.html:233](/home/liam/aw-claude-assaying-boyle-1014/public/strata/most-numbers-begin-with-one/index.html:233): before and after: ```diff @@ -232,1 +233,1 @@ -

Used as a verdict, it lies. The most common abuse is the viral claim that a candidate's vote counts "violate Benford's law, therefore fraud." They almost always do violate it \u2014 and that proves nothing, because precinct vote totals are exactly the kind of bounded, narrow-range data Benford does not govern. Precincts are sized to hold roughly the same number of voters, so the counts cluster in a band rather than spanning decades; a candidate with ~500 votes per precinct will show a glut of leading 4s and 5s with no fraud anywhere. In 2011 the political scientists Deckert, Myagkov, and Ordeshook showed in Political Analysis that the first-digit test is "problematical at best as a forensic tool when applied to elections" \u2014 it both misses real fraud and manufactures false alarms. A law that says most numbers begin with one can tell you an invoice file looks invented. It cannot tell you who won an election.

+

Used as a verdict, it lies. The most common abuse is the viral claim that a candidate's vote counts "violate Benford's law, therefore fraud." They almost always do violate it \u2014 and that proves nothing, because precinct vote totals are exactly the kind of bounded, narrow-range data Benford does not govern. Precincts are sized to hold roughly the same number of voters, so the counts cluster in a band rather than spanning decades; a candidate with ~500 votes per precinct will show a glut of leading 4s and 5s with no fraud anywhere. In 2011 the political scientists Deckert, Myagkov, and Ordeshook showed in Political Analysis that the first-digit test is "problematical at best as a forensic tool when applied to elections" \u2014 it both misses real fraud and manufactures false alarms. A law that says one is the most frequent first digit can tell you an invoice file looks invented. It cannot tell you who won an election.

``` - [public/strata/most-numbers-begin-with-one/index.html:239](/home/liam/aw-claude-assaying-boyle-1014/public/strata/most-numbers-begin-with-one/index.html:239): before and after: ```diff @@ -238,2 +239,2 @@ -

What is checked vs. what is cited. The law and its consequences are recomputed from first principles. The datasets are real, published figures, committed as-is: country populations, surface areas and GDP from the World Bank (2022, CC-BY 4.0), and the NIST CODATA reference constants (public domain). The page does not claim those figures are exact truth \u2014 only that this committed snapshot has the leading-digit distribution shown, reproducibly. Histograms on the page match the verifier exactly because rounding the snapshot to six significant figures cannot change a leading digit.

-

Sources: Newcomb, Note on the Frequency of Use of the Different Digits in Natural Numbers (American Journal of Mathematics 4, 1881, pp. 39–40); Benford, The Law of Anomalous Numbers (Proc. American Philosophical Society 78, 1938, pp. 551–572; 20,229 observations); Pinkham, On the Distribution of First Significant Digits (Ann. Math. Statist. 32, 1961 \u2014 scale invariance ⇒ Benford); Hill, A Statistical Derivation of the Significant-Digit Law (Statistical Science 10, 1995) and Base-Invariance Implies Benford's Law (Proc. AMS 123, 1995); Nigrini, Benford's Law: Applications for Forensic Accounting (2012); Deckert, Myagkov & Ordeshook, Benford's Law and the Detection of Election Fraud (Political Analysis 19, 2011, pp. 245–268).

+

What is checked vs. what is cited. The law and its consequences are recomputed from first principles. The datasets are numerical snapshots extracted from published figures: country populations, surface areas and GDP from the World Bank (2022, CC-BY 4.0), and the NIST CODATA reference constants (public domain). The page does not claim those figures are exact truth \u2014 only that this committed snapshot has the leading-digit distribution shown, reproducibly. The compacted snapshot's first-digit counts have been checked against the full numerical snapshots and match for all four datasets. Rounding can change a leading digit; that equality is a checked fact about these values, not a general guarantee.

+

Sources: Newcomb, Note on the Frequency of Use of the Different Digits in Natural Numbers (American Journal of Mathematics 4, 1881, pp. 39–40); Benford, The Law of Anomalous Numbers (Proc. American Philosophical Society 78, 1938, pp. 551–572; 20,229 observations); Pinkham, On the Distribution of First Significant Digits (Ann. Math. Statist. 32, 1961; a flawed scale-invariance argument); Hill, A Statistical Derivation of the Significant-Digit Law (Statistical Science 10, 1995) and Base-Invariance Implies Benford's Law (Proc. AMS 123, 1995); Nigrini, Benford's Law: Applications for Forensic Accounting (2012); Deckert, Myagkov & Ordeshook, Benford's Law and the Detection of Election Fraud (Political Analysis 19, 2011, pp. 245–268).

``` - [public/strata/most-numbers-begin-with-one/index.html:266](/home/liam/aw-claude-assaying-boyle-1014/public/strata/most-numbers-begin-with-one/index.html:266): before and after: ```diff @@ -265,0 +266,1 @@ + const ben = benId ? $(benId) : null; ``` - [public/strata/most-numbers-begin-with-one/index.html:270](/home/liam/aw-claude-assaying-boyle-1014/public/strata/most-numbers-begin-with-one/index.html:270): before and after: ```diff @@ -268,5 +270,10 @@ - if (benId) { - // place a Benford dot at the centre of each column - const ben = $(benId); - ben.innerHTML = BEN.map((b, i) => - ``).join(''); + if (ben) { + // The overlay shares the bars' plotting area, including their baseline. + const bounds = host.getBoundingClientRect(); + const points = BEN.map((b, i) => { + const col = host.children[i].getBoundingClientRect(); + return [(col.left + col.width / 2 - bounds.left) / bounds.width * 100, b / top * 100]; + }); + host.appendChild(ben); + ben.innerHTML = `` + + points.map(([x, y]) => ``).join(''); ``` - [public/strata/most-numbers-begin-with-one/index.html:323](/home/liam/aw-claude-assaying-boyle-1014/public/strata/most-numbers-begin-with-one/index.html:323): before and after: ```diff @@ -316,1 +323,1 @@ - : `Pulled from real life, never arranged to fit \u2014 and the bars track the dashed Benford curve closely (P(1) = ${(p[0] * 100).toFixed(1)}% vs 30.1%). The data spans ${orders.toFixed(1)} orders of magnitude, which is exactly the condition the law needs.`; + : `Pulled from real life, never arranged to fit \u2014 and the bars track the dashed Benford curve closely (P(1) = ${(p[0] * 100).toFixed(1)}% vs 30.1%). The data spans ${orders.toFixed(1)} orders of magnitude, but that range alone neither guarantees the law nor is necessary for it.`; ``` - [public/strata/most-numbers-begin-with-one/index.html:405](/home/liam/aw-claude-assaying-boyle-1014/public/strata/most-numbers-begin-with-one/index.html:405): before and after: ```diff @@ -398,0 +405,1 @@ + $('seqLimit').textContent = isLaw ? 'total variation, → 0' : 'total variation, no convergence to 0'; ``` - [public/strata/most-numbers-begin-with-one/index.html:414](/home/liam/aw-claude-assaying-boyle-1014/public/strata/most-numbers-begin-with-one/index.html:414): before and after: ```diff @@ -406,0 +414,1 @@ +window.addEventListener('resize', () => { renderData(); renderSeq(); }); ``` - [public/strata/most-numbers-begin-with-one/index.html:422](/home/liam/aw-claude-assaying-boyle-1014/public/strata/most-numbers-begin-with-one/index.html:422): before: no Corrections note. After: thirteen exact dated paragraphs listed above, followed by primary-source links. - [public/strata/most-numbers-begin-with-one/index.html:463](/home/liam/aw-claude-assaying-boyle-1014/public/strata/most-numbers-begin-with-one/index.html:463): before and after: ```diff @@ -435,1 +463,1 @@ -
  • The Law Even Monkeys ObeyTwo heavy-skew laws of real-world numbers, both born of multiplicative structure.
    • +
  • The Law Even Monkeys ObeyTwo skew laws of real-world numbers.
    • ``` - [public/strata/most-numbers-begin-with-one/index.html:465](/home/liam/aw-claude-assaying-boyle-1014/public/strata/most-numbers-begin-with-one/index.html:465): before and after: ```diff @@ -437,1 +465,1 @@ -
  • The Tanks That Counted ThemselvesBoth read a truth off numbers that were never meant to carry it \u2014 a serial number quietly encodes a production run; a first digit quietly encodes whether a dataset grew…
    • +
  • The Tanks That Counted ThemselvesSerial numbers can inform a production estimate under a numbering model; first digits can test a proposed distribution, but cannot establish a dataset's growth history.
    • ``` - [src/content/strata/most-numbers-begin-with-one.md:2](/home/liam/aw-claude-assaying-boyle-1014/src/content/strata/most-numbers-begin-with-one.md:2): before and after: ```diff @@ -2,1 +2,1 @@ -title: "Most Numbers Begin With One" +title: "Benford's Law: One Leads Most Often" ``` - [src/content/strata/most-numbers-begin-with-one.md:5](/home/liam/aw-claude-assaying-boyle-1014/src/content/strata/most-numbers-begin-with-one.md:5): before and after: ```diff @@ -5,1 +5,1 @@ -dek: "Take a long list of real-world numbers (country populations, land areas, the physical constants) and look only at the first digit of each. A 1 turns up about 30% of the time; a 9, less than 5%. That lopsidedness is Benford's law, P(d) = log₁₀(1 + 1/d), and it isn't a quirk of any one dataset: it is the only leading-digit pattern that survives a change of units. An immersive page proves it three ways (on four real datasets, on the powers of two by exact integer arithmetic, and on the log-circle where scale-invariance forces it), then shows exactly where it fails, including the place it is most often abused: the claim that a vote count 'violates Benford, therefore fraud.' Every figure recomputed in your browser and offline (52/52 + a 14-check browser pass)." +dek: "Take a long list of real-world numbers (country populations, surface areas, the physical constants) and look only at the first digit of each. A 1 turns up about 30% of the time; a 9, less than 5%. That lopsidedness is Benford's law, P(d) = log₁₀(1 + 1/d), and it isn't a quirk of any one dataset: it is the only leading-digit pattern that survives a change of units. An immersive page proves it three ways (on four real datasets, on the powers of two by exact integer arithmetic, and on the log-circle where scale-invariance forces it), then shows exactly where it fails, including the place it is most often abused: the claim that a vote count 'violates Benford, therefore fraud.' Every figure recomputed in your browser and offline (52/52 + a 26-check browser pass)." ``` - [src/content/strata/most-numbers-begin-with-one.md:13](/home/liam/aw-claude-assaying-boyle-1014/src/content/strata/most-numbers-begin-with-one.md:13): before and after: ```diff @@ -13,1 +13,1 @@ - note: "Two heavy-skew laws of real-world numbers, both born of multiplicative structure. Zipf ranks words and finds frequency ∝ 1/rank; Benford reads first digits and finds P(d) = log₁₀(1+1/d). Both are what you get when quantities are spread evenly across orders of magnitude rather than added up." + note: "Two skew laws of real-world numbers. Zipf ranks words and finds frequency ∝ 1/rank; Benford reads first digits and finds P(d) = log₁₀(1+1/d). The first describes a rank-frequency relation; the second describes uniformly distributed fractional logarithms. They are different distributional statements." ``` - [src/content/strata/most-numbers-begin-with-one.md:17](/home/liam/aw-claude-assaying-boyle-1014/src/content/strata/most-numbers-begin-with-one.md:17): before and after: ```diff @@ -17,1 +17,1 @@ - note: "Both read a truth off numbers that were never meant to carry it \u2014 a serial number quietly encodes a production run; a first digit quietly encodes whether a dataset grew multiplicatively. And both end on the same honesty move: naming precisely the assumptions under which the inference holds, and where it silently breaks." + note: "Both find information in numbers: serial numbers can inform a production estimate under a numbering model; first digits can test a proposed distribution, but cannot establish a dataset's growth history. And both end on the same honesty move: naming precisely the assumptions under which the inference holds, and where it silently breaks." ``` - [src/content/strata/most-numbers-begin-with-one.md:21](/home/liam/aw-claude-assaying-boyle-1014/src/content/strata/most-numbers-begin-with-one.md:21): before and after: ```diff @@ -21,3 +21,3 @@ -Take a long list of real-world numbers \u2014 the population and the land area of -every country, the size of every economy, the 355 constants in the international -physics reference table \u2014 and throw away everything but the **first digit** of each. You would +Take four published numerical snapshots: populations for 217 countries and +economies, surface areas for 215, GDP for 209, and the 355 constants in the +international physics reference table. Throw away everything but the **first digit** of each. You would ``` - [src/content/strata/most-numbers-begin-with-one.md:40](/home/liam/aw-claude-assaying-boyle-1014/src/content/strata/most-numbers-begin-with-one.md:40): before and after: ```diff @@ -40,1 +40,1 @@ -the distribution refuse to move; and the honest boundary \u2014 where Benford breaks, +the sample proportions change while the ideal Benford law stays invariant; and the honest boundary \u2014 where Benford breaks, ``` - [src/content/strata/most-numbers-begin-with-one.md:45](/home/liam/aw-claude-assaying-boyle-1014/src/content/strata/most-numbers-begin-with-one.md:45): before: Markdown ended after the verifier paragraph. After: thirteen mirrored correction paragraphs listed above. - [research/the-first-digit/README.md:3](/home/liam/aw-claude-assaying-boyle-1014/research/the-first-digit/README.md:3): before and after: ```diff @@ -3,1 +3,1 @@ -Lab notebook for the stratum **"Most Numbers Begin With One"** +Lab notebook for the stratum **"Benford's Law: One Leads Most Often"** ``` - [research/the-first-digit/README.md:16](/home/liam/aw-claude-assaying-boyle-1014/research/the-first-digit/README.md:16): before and after: ```diff @@ -16,1 +16,1 @@ -- `verify-page.mjs` \u2014 Playwright check of the immersive page itself, **14/14**: +- `verify-page.mjs` \u2014 Playwright check of the immersive page itself, **26/26**: ``` - [research/the-first-digit/README.md:23](/home/liam/aw-claude-assaying-boyle-1014/research/the-first-digit/README.md:23): before and after: ```diff @@ -23,1 +23,1 @@ - real sovereign states only (aggregates filtered out). **CC-BY 4.0.** + countries and economies (aggregates filtered out): 217 population, 215 surface-area and 209 GDP entries. **CC-BY 4.0.** ``` - [research/the-first-digit/README.md:26](/home/liam/aw-claude-assaying-boyle-1014/research/the-first-digit/README.md:26): before and after: ```diff @@ -26,0 +26,4 @@ + `codata-2022-source.txt` preserves the source table retrieved 2026-10-14. + `build.mjs` retains the exponents after ellipses and compacts values to eight + significant figures; it updates the raw constants, compact snapshot, inline + DATA statement and this layer's published verifier. The article is hand-authored. ``` - [research/the-first-digit/README.md:32](/home/liam/aw-claude-assaying-boyle-1014/research/the-first-digit/README.md:32): before and after: ```diff @@ -28,2 +32,2 @@ - (rounded to ≥6 significant figures, which cannot change a leading digit, so - the page's histograms match the verifier's exactly). + (compacted numerical values; all four first-digit histograms are checked against + the full snapshots. Rounding is not guaranteed to preserve a first digit). ``` - [research/the-first-digit/README.md:38](/home/liam/aw-claude-assaying-boyle-1014/research/the-first-digit/README.md:38): before and after: ```diff @@ -34,0 +38,1 @@ +node research/the-first-digit/build.mjs --check # source, extraction, inline data and check mirror ``` - [research/the-first-digit/README.md:40](/home/liam/aw-claude-assaying-boyle-1014/research/the-first-digit/README.md:40): before and after: ```diff @@ -35,1 +40,1 @@ -node research/the-first-digit/verify-page.mjs # 14/14, needs Playwright +node research/the-first-digit/verify-page.mjs # 26/26, needs Playwright ``` - [research/the-first-digit/README.md:46](/home/liam/aw-claude-assaying-boyle-1014/research/the-first-digit/README.md:46): before and after: ```diff @@ -41,1 +46,1 @@ -real published figures, committed as-is; the page does not claim they are exact +numerical extractions of published figures; the page does not claim they are exact ``` - [research/the-first-digit/og.html:29](/home/liam/aw-claude-assaying-boyle-1014/research/the-first-digit/og.html:29): before and after: ```diff @@ -29,2 +29,2 @@ -

    Most Numbers Begin With One

    -
    Take real numbers \u2014 populations, land areas, the physical constants \u2014 and keep only the first digit. A 1 leads about 30% of the time. It's the one leading-digit law that survives a change of units.
    +

    Benford's Law: One Leads Most Often

    +
    Take real numbers (populations, surface areas, the physical constants) and keep only the first digit. A 1 leads about 30% of the time. It's the one leading-digit law that survives a change of units.
    ``` - [research/the-first-digit/verify.mjs:2](/home/liam/aw-claude-assaying-boyle-1014/research/the-first-digit/verify.mjs:2): before and after: ```diff @@ -2,1 +2,1 @@ -// Offline verifier for the stratum "Most Numbers Begin With One" (Benford's law). +// Offline verifier for the stratum "Benford's Law: One Leads Most Often". ``` - [research/the-first-digit/verify.mjs:74](/home/liam/aw-claude-assaying-boyle-1014/research/the-first-digit/verify.mjs:74): before and after: ```diff @@ -74,1 +74,1 @@ -console.log('\nMost Numbers Begin With One \u2014 offline verification (Benford\'s law)\n'); +console.log('\nBenford\'s Law: One Leads Most Often - offline verification\n'); ``` - [research/the-first-digit/verify-page.mjs:5](/home/liam/aw-claude-assaying-boyle-1014/research/the-first-digit/verify-page.mjs:5): before and after: ```diff @@ -5,1 +5,2 @@ -import pw from '/opt/node22/lib/node_modules/playwright/index.js'; +import pw from 'playwright'; +import { readFileSync } from 'node:fs'; ``` - [research/the-first-digit/verify-page.mjs:18](/home/liam/aw-claude-assaying-boyle-1014/research/the-first-digit/verify-page.mjs:18): before and after: ```diff @@ -17,0 +18,1 @@ +await page.route(/^https?:/, (route) => route.abort()); ``` - [research/the-first-digit/verify-page.mjs:26](/home/liam/aw-claude-assaying-boyle-1014/research/the-first-digit/verify-page.mjs:26): before and after: ```diff @@ -24,1 +26,1 @@ -console.log('\nMost Numbers Begin With One \u2014 browser check\n'); +console.log('\nBenford\'s Law: One Leads Most Often - browser check\n'); ``` - [research/the-first-digit/verify-page.mjs:47](/home/liam/aw-claude-assaying-boyle-1014/research/the-first-digit/verify-page.mjs:47): before and after: ```diff @@ -45,1 +47,1 @@ -// scale slider: rescaling barely moves P(1) (back on populations) +// Rescaling a finite sample can change its first-digit proportions. ``` - [research/the-first-digit/verify-page.mjs:49](/home/liam/aw-claude-assaying-boyle-1014/research/the-first-digit/verify-page.mjs:49): before and after: ```diff @@ -47,5 +49,6 @@ -await page.waitForTimeout(100); -await page.locator('#scaleSld').fill('2400'); // ×10^2.4 ≈ ×251 -await page.waitForTimeout(150); -const cP1 = parseFloat(await page.locator('#cP1').textContent()); -ok('scale-invariance: P(1) stays in [25,36]% after ×~251', cP1 >= 25 && cP1 <= 36, cP1 + '%'); +await page.locator('#scaleSld').fill('-140'); +ok('population P(1) changes to 22.6% at c = 10^(-0.14)', + await page.locator('#cP1').textContent() === '22.6%'); +await page.locator('#scaleSld').fill('1000'); +ok('multiplication by ten preserves population P(1) = 30.9%', + await page.locator('#cP1').textContent() === '30.9%'); ``` - [research/the-first-digit/verify-page.mjs:62](/home/liam/aw-claude-assaying-boyle-1014/research/the-first-digit/verify-page.mjs:62): before and after: ```diff @@ -59,0 +62,3 @@ +ok('powers-of-two limit label tends to zero', + await page.locator('#seqLimit').textContent() === 'total variation, → 0'); + ``` - [research/the-first-digit/verify-page.mjs:69](/home/liam/aw-claude-assaying-boyle-1014/research/the-first-digit/verify-page.mjs:69): before and after: ```diff @@ -63,1 +69,1 @@ -ok('counting numbers do NOT converge (TVD notably larger)', tvd2 > tvd + 0.02, 'TVD=' + tvd2); +ok('counting sample is farther from Benford at N=4000', tvd2 > tvd + 0.02, 'TVD=' + tvd2); ``` - [research/the-first-digit/verify-page.mjs:71](/home/liam/aw-claude-assaying-boyle-1014/research/the-first-digit/verify-page.mjs:71): before and after: ```diff @@ -65,0 +71,3 @@ + +ok('counting limit label does not promise convergence to zero', + await page.locator('#seqLimit').textContent() === 'total variation, no convergence to 0'); ``` - [research/the-first-digit/verify-page.mjs:80](/home/liam/aw-claude-assaying-boyle-1014/research/the-first-digit/verify-page.mjs:80): before and after: ```diff @@ -71,0 +80,3 @@ +ok('factorial limit label returns to convergence', + await page.locator('#seqLimit').textContent() === 'total variation, → 0'); + ``` - [research/the-first-digit/verify-page.mjs:85](/home/liam/aw-claude-assaying-boyle-1014/research/the-first-digit/verify-page.mjs:85): before and after: ```diff @@ -73,0 +85,52 @@ + +// Compare actual bar readouts against full source snapshots, independently of DATA. +const snapshots = [ + ['Country populations', 'population-2022.json'], + ['Country surface areas', 'surface-area-2022.json'], + ['National GDP', 'gdp-usd-2022.json'], + ['Physical constants', 'codata-constants.json'], +]; +for (const [label, file] of snapshots) { + const rows = JSON.parse(readFileSync(path.join(HERE, 'data', file), 'utf8')); + const counts = Array(9).fill(0); + for (const row of rows) counts[Number(Math.abs(row.value).toExponential()[0]) - 1]++; + await page.locator('#pickData button', { hasText: label }).click(); + const shown = await page.locator('#dataHist .bar').evaluateAll((bars) => bars.map((bar) => bar.title)); + ok(`${label}: compacted histogram equals full-snapshot counts`, + shown.every((text, i) => text === `digit ${i + 1}: ${(counts[i] / rows.length * 100).toFixed(1)}%`)); +} + +// Measure reference positions against the mathematical probabilities and actual columns. +async function overlayMatches(histId, benId) { + return page.evaluate(({ histId, benId }) => { + const host = document.getElementById(histId); + const overlay = document.getElementById(benId); + const bounds = host.getBoundingClientRect(); + const columns = [...host.querySelectorAll('.col')]; + const dots = [...overlay.querySelectorAll('.bendot')]; + const probabilities = Array.from({ length: 9 }, (_, i) => Math.log10((i + 2) / (i + 1))); + // Recover the active plot scale, then compare geometry with the Benford formula. + const proportions = histId === 'dataHist' + ? propsOf(hist9(DATA[curData].v)) : propsOf(seqPrefix[+document.getElementById('seqSld').value]); + const top = Math.max(...proportions, ...probabilities) * 1.08; + const line = overlay.querySelector('polyline'); + return overlay.parentElement === host && overlay.clientHeight === host.clientHeight && + dots.length === 9 && line?.points.numberOfItems === 9 && line.getAttribute('stroke-dasharray') === '5 4' && + dots.every((dot, i) => { + const rect = dot.getBoundingClientRect(); + const col = columns[i].getBoundingClientRect(); + const expectedY = bounds.top + host.clientHeight * (1 - probabilities[i] / top); + const point = line.points.getItem(i); + return Math.abs(rect.top + rect.height / 2 - expectedY) < 0.8 && + Math.abs(rect.left + rect.width / 2 - col.left - col.width / 2) < 0.8 && + Math.abs(point.y - (1 - probabilities[i] / top) * 100) < 0.001; + }); + }, { histId, benId }); +} +ok('data reference follows Benford probabilities at column centres', await overlayMatches('dataHist', 'dataBen')); +ok('sequence reference follows Benford probabilities at column centres', await overlayMatches('seqHist', 'seqBen')); +await page.setViewportSize({ width: 390, height: 844 }); +await page.waitForTimeout(100); +ok('both references stay aligned at mobile width', + await overlayMatches('dataHist', 'dataBen') && await overlayMatches('seqHist', 'seqBen')); +ok('no page errors after all interactions', errors.length === 0, errors.join(' | ') || 'clean'); ``` - [public/checks/research/the-first-digit/verify.mjs:2](/home/liam/aw-claude-assaying-boyle-1014/public/checks/research/the-first-digit/verify.mjs:2): before and after: ```diff @@ -2,1 +2,1 @@ -// Offline verifier for the stratum "Most Numbers Begin With One" (Benford's law). +// Offline verifier for the stratum "Benford's Law: One Leads Most Often". ``` - [public/checks/research/the-first-digit/verify.mjs:74](/home/liam/aw-claude-assaying-boyle-1014/public/checks/research/the-first-digit/verify.mjs:74): before and after: ```diff @@ -74,1 +74,1 @@ -console.log('\nMost Numbers Begin With One \u2014 offline verification (Benford\'s law)\n'); +console.log('\nBenford\'s Law: One Leads Most Often - offline verification\n'); ``` ### CODATA values and regenerated copies All replacements below are from the matching named row in the [NIST 2022 table](https://physics.nist.gov/cuu/Constants/Table/allascii.txt), retained in `research/the-first-digit/data/codata-2022-source.txt`. The parser preserves exponents after ellipses and uses the printed decimals as finite numerical approximations. The absolute-value convention is unchanged. Every raw change is at [research/the-first-digit/data/codata-constants.json:1](/home/liam/aw-claude-assaying-boyle-1014/research/the-first-digit/data/codata-constants.json:1). Every compact change occurs in both [research/the-first-digit/data/page-snapshot.json:1](/home/liam/aw-claude-assaying-boyle-1014/research/the-first-digit/data/page-snapshot.json:1) and [public/strata/most-numbers-begin-with-one/index.html:246](/home/liam/aw-claude-assaying-boyle-1014/public/strata/most-numbers-begin-with-one/index.html:246). Row numbers below locate the source table. | Constant / source line | Raw before → after | Compact and inline before → after | | --- | --- | --- | | atomic unit of action (source line 34) | `1.054571817` → `1.054571817e-34` | `1.0545718` → `1.0545718e-34` | | Boltzmann constant in eV/K (source line 63) | `8.617333262` → `8.617333262e-05` | `8.6173333` → `8.6173333e-05` | | Boltzmann constant in Hz/K (source line 64) | `2.083661912` → `20836619120` | `2.0836619` → `20836619000` | | conductance quantum (source line 69) | `7.748091729` → `7.748091729e-05` | `7.7480917` → `7.7480917e-05` | | electron volt-hertz relationship (source line 127) | `2.417989242` → `241798924200000` | `2.4179892` → `241798920000000` | | electron volt-inverse meter relationship (source line 128) | `8.065543937` → `806554.3937` | `8.0655439` → `806554.39` | | electron volt-kelvin relationship (source line 130) | `1.160451812` → `11604.51812` | `1.1604518` → `11604.518` | | electron volt-kilogram relationship (source line 131) | `1.782661921` → `1.782661921e-36` | `1.7826619` → `1.7826619e-36` | | elementary charge over h-bar (source line 133) | `1.519267447` → `1519267447000000` | `1.5192674` → `1519267400000000` | | first radiation constant (source line 137) | `3.741771852` → `3.741771852e-16` | `3.7417719` → `3.7417719e-16` | | first radiation constant for spectral radiance (source line 138) | `1.191042972` → `1.191042972e-16` | `1.191043` → `1.191043e-16` | | hertz-electron volt relationship (source line 162) | `4.135667696` → `4.135667696e-15` | `4.1356677` → `4.1356677e-15` | | hertz-inverse meter relationship (source line 164) | `3.335640951` → `3.335640951e-09` | `3.335641` → `3.335641e-09` | | hertz-kelvin relationship (source line 166) | `4.799243073` → `4.799243073e-11` | `4.7992431` → `4.7992431e-11` | | hertz-kilogram relationship (source line 167) | `7.372497323` → `7.372497323e-51` | `7.3724973` → `7.3724973e-51` | | inverse meter-electron volt relationship (source line 171) | `1.239841984` → `1.239841984e-06` | `1.239842` → `1.239842e-06` | | inverse meter-joule relationship (source line 174) | `1.986445857` → `1.986445857e-25` | `1.9864459` → `1.9864459e-25` | | inverse meter-kelvin relationship (source line 175) | `1.438776877` → `0.01438776877` | `1.4387769` → `0.014387769` | | inverse meter-kilogram relationship (source line 176) | `2.210219094` → `2.210219094e-42` | `2.2102191` → `2.2102191e-42` | | Josephson constant (source line 74) | `483597.8484` → `483597848400000` | `483597.85` → `483597850000000` | | joule-electron volt relationship (source line 180) | `6.241509074` → `6241509074000000000` | `6.2415091` → `6241509100000000000` | | joule-hertz relationship (source line 182) | `1.509190179` → `1.509190179e+33` | `1.5091902` → `1.5091902e+33` | | joule-inverse meter relationship (source line 183) | `5.034116567` → `5.034116567e+24` | `5.0341166` → `5.0341166e+24` | | joule-kelvin relationship (source line 184) | `7.242970516` → `7.242970516e+22` | `7.2429705` → `7.2429705e+22` | | joule-kilogram relationship (source line 185) | `1.112650056` → `1.112650056e-17` | `1.1126501` → `1.1126501e-17` | | kelvin-electron volt relationship (source line 187) | `8.617333262` → `8.617333262e-05` | `8.6173333` → `8.6173333e-05` | | kelvin-hertz relationship (source line 189) | `2.083661912` → `20836619120` | `2.0836619` → `20836619000` | | kelvin-kilogram relationship (source line 192) | `1.536179187` → `1.536179187e-40` | `1.5361792` → `1.5361792e-40` | | kilogram-electron volt relationship (source line 194) | `5.609588603` → `5.609588603e+35` | `5.6095886` → `5.6095886e+35` | | kilogram-hertz relationship (source line 196) | `1.356392489` → `1.356392489e+50` | `1.3563925` → `1.3563925e+50` | | kilogram-inverse meter relationship (source line 197) | `4.524438335` → `4.524438335e+41` | `4.5244383` → `4.5244383e+41` | | kilogram-joule relationship (source line 198) | `8.987551787` → `89875517870000000` | `8.9875518` → `89875518000000000` | | kilogram-kelvin relationship (source line 199) | `6.50965726` → `6.50965726e+39` | `6.5096573` → `6.5096573e+39` | | Loschmidt constant (273.15 K, 100 kPa) (source line 202) | `2.651645804` → `2.651645804e+25` | `2.6516458` → `2.6516458e+25` | | Loschmidt constant (273.15 K, 101.325 kPa) (source line 203) | `2.686780111` → `2.686780111e+25` | `2.6867801` → `2.6867801e+25` | | mag. flux quantum (source line 205) | `2.067833848` → `2.067833848e-15` | `2.0678338` → `2.0678338e-15` | | molar Planck constant (source line 209) | `3.990312712` → `3.990312712e-10` | `3.9903127` → `3.9903127e-10` | | molar volume of ideal gas (273.15 K, 100 kPa) (source line 210) | `22.71095464` → `0.02271095464` | `22.710955` → `0.022710955` | | molar volume of ideal gas (273.15 K, 101.325 kPa) (source line 211) | `22.41396954` → `0.02241396954` | `22.41397` → `0.02241397` | | natural unit of action (source line 230) | `1.054571817` → `1.054571817e-34` | `1.0545718` → `1.0545718e-34` | | natural unit of action in eV s (source line 231) | `6.582119569` → `6.582119569e-16` | `6.5821196` → `6.5821196e-16` | | Planck constant in eV/Hz (source line 272) | `4.135667696` → `4.135667696e-15` | `4.1356677` → `4.1356677e-15` | | reduced Planck constant (source line 304) | `1.054571817` → `1.054571817e-34` | `1.0545718` → `1.0545718e-34` | | reduced Planck constant in eV s (source line 305) | `6.582119569` → `6.582119569e-16` | `6.5821196` → `6.5821196e-16` | | second radiation constant (source line 315) | `1.438776877` → `0.01438776877` | `1.4387769` → `0.014387769` | | Stefan-Boltzmann constant (source line 334) | `5.670374419` → `5.670374419e-08` | `5.6703744` → `5.6703744e-08` | | Wien frequency displacement law constant (source line 364) | `5.878925757` → `58789257570` | `5.8789258` → `58789258000` | | Wien wavelength displacement law constant (source line 365) | `2.897771955` → `0.002897771955` | `2.897772` → `0.002897772` | The four full-snapshot and compact-snapshot first-digit histograms remain exactly equal. The 48 exponent changes leave all four digit histograms unchanged. The constants range readout changes from 98.5 to 114.3 decades; the source-derived value is 114.33752245242302. No claim that the ranges are unchanged is made. ### New layer files - [research/the-first-digit/data/codata-2022-source.txt:1](/home/liam/aw-claude-assaying-boyle-1014/research/the-first-digit/data/codata-2022-source.txt:1): before: absent. After: the 40,801-byte 2022 NIST table retrieved 2026-10-14, SHA-256 `77fb90e66c40db3e6eb16630bc9c88e4c7c8beddbe5e71be406f2f26e3f67e67`. This small source snapshot is retained for reproducibility; no bulk dataset or archive was downloaded. - [research/the-first-digit/build.mjs:1](/home/liam/aw-claude-assaying-boyle-1014/research/the-first-digit/build.mjs:1): before: no layer data rebuild script. After: a parser for all 355 named NIST rows, preserving exponents after ellipses; a rebuild of the raw constants, compact constants, inline DATA statement, and this layer's published verifier; `--check` compares those four files without writing. The article prose remains hand-authored. - [assay/records/most-numbers-begin-with-one.md:1](/home/liam/aw-claude-assaying-boyle-1014/assay/records/most-numbers-begin-with-one.md:1): before: absent. After: this 144-claim record, 39 findings, all 40 triage outcomes, exact dated lines, complete edit ledger, commands/results and unfinished work. ## Commands and results - Read-only discovery: `pwd`, scoped `rg --files`, `rg -n`, `cat`, `sed`, Python JSON inspection, `git rev-parse HEAD`, scoped `git status --short`, and `git log -4 --format=... --date=short -- public/strata/most-numbers-begin-with-one/index.html`. HEAD matched the requested `f5e5f84f34`; this layer's editable files were clean before the repair. No AGENTS.md was found. No layer page generator existed; the research directory contained the authored page's data, verifiers and a separate OG renderer. The unrelated `research/benford` directory was inspected only to rule out a generator for this layer and was not edited. - Read primary papers with the web tool: Berger and Hill's primer; Hill on range conditions; Benford 1938; Berger and Hill section 4.2 and Theorem 4.20; Hill's 1995 abstract; Newman equation (4) and Appendix A; Deckert et al. abstract and p. 247; World Bank indicator definition; NIST table. The CiNii browser open failed, then direct retrieval succeeded. No request included an email address or personal details. - Python `urllib.request.urlopen` downloads with a 2,000,000-byte cap: World Bank indicator JSON succeeded (596 bytes), CiNii catalogue succeeded (51,712 bytes), and the initial NIST request returned HTTP 403. `curl --fail --location --max-time 30 --max-filesize 2000000 -A 'Mozilla/5.0' https://physics.nist.gov/cuu/Constants/Table/allascii.txt -o /tmp/aw-most-numbers-1014/nist.txt` then succeeded (40,801 bytes). Total successful response bodies downloaded: 93,109 bytes. No large download, OCR, PDF rendering or media generation ran. - Inline Node recomputation of the original DATA statement and exact sequence generators: four raw/compact histograms matched; population P(1)=30.9% at c=1 and 22.6% at c=10^(-0.14), with c=10 restoring 30.9%; the full slider sweeps and all listed sequence distances reproduced the triage. The original Chromium DOM probe found all nine data reference dots at the same y coordinate. No screenshot was rendered or saved. - The first Python integer/float comparison of NIST values reported 49 textual numeric differences; one was a large-integer floating-point representation comparison. A JavaScript Number comparison of all 355 rows established exactly 48 value changes, all in source rows with ellipses. Only those 48 numerical values were repaired. - `python3 /tmp/aw-most-numbers-1014/edit-layer.py`: the initial exact-match guard stopped before writing when it found three land-area copies rather than two. After accounting for all three, the command applied 43 scoped text replacements and the 13 mirrored correction paragraphs. Later small wording and browser-count edits are included in the ledger. - `node research/the-first-digit/build.mjs`: exit 0; refreshed 355 CODATA values and exactly four layer files. It regenerated the published verifier copy rather than editing that copy directly. - `node research/the-first-digit/build.mjs --check`: exit 0; all 355 parsed values and four regenerated files match their sources. - `node research/the-first-digit/verify.mjs`: exit 0, `PASS 52/52 checks`. The four counts, P(1), chi-square values and exact-sequence checks pass; the corrected constants span 114.3 decades. - `node research/the-first-digit/verify-page.mjs`: exit 0, `PASS 26/26 browser checks`. Correct scaling behavior, sequence-dependent limit labels, four dataset comparisons, dashed overlay positions at desktop and mobile widths, and no page errors. The original 14-check source had a hard-coded unavailable Playwright path; it now imports the installed `playwright` package. No install was needed. - Inline Node repair checks: all nine listed MINOR entries' shared authored sentences and X24's hallmark are byte-for-byte unchanged; all 13 HTML/Markdown correction pairs match; the Corrections note is above the placard; all four before/after full and compact histograms agree; the rounded-invoice counterexample has TVD=0.000015724144730465406. - Scoped `git diff --check`: initially reported one extra blank line at the end of the Markdown; removed it. Final scoped run exited 0 with no whitespace errors. - `node scripts/check-claims-pass.mjs`: exit 1 only for eight other layer records: `any-way-you-fall`, `helium-voice`, `island-biogeography`, `khinchin`, `let-them-eat-cake`, `the-bread-with-no-name`, `the-drop-that-shatters-itself`, and `the-edge-of-the-bow`. Each reported malformed findings. This layer has no gate errors; the other records were left untouched as instructed. - Direct `checkRecord('most-numbers-begin-with-one', ..., new Set())` from `scripts/check-claims-pass.mjs`: exit 0, `errs: []`, checked 144, confirmed 105, wrong 38 (29 WRONG plus nine MINOR), stale 1, open 0. ## Not done - X7's dek-derived land-area wording has the directed [fixed] disposition: share card rebuilt at landing from the corrected dek. Both current image files were inspected and still show the old wording; no generator was run. The directing instance should confirm that the landing output also refreshes the shared WebP thumbnail. - X2's old majority title in the PNG and WebP remains open board notes, with OG REGEN NEEDED in the repair report because the title is separate from the dek. The four authored links in other layers also remain open board notes. Their paths, replacement labels and board-note requests are in research/assay-line/cited-1014b/most-numbers-begin-with-one/repair-report.md. - X13, X21 and X28-X34 remain unchanged as directed, recorded as MINOR with [declined] findings and the triage's tier-C reasons. X24 was rejected with the hallmark-definition evidence and left unchanged. - No shared placard/assay-fold/search/OG/site rebuild was run. The published hallmark can advance through the normal landing workflow; the user prohibited running the mutating placard command. No other layer, shared file, git-writing command, package installation, full site build, OEIS derive script, film, audio or image was changed.